1. Field of the Invention
The present invention relates to an apparatus and method for orthogonal frequency division multiplexing (OFDM) channel equalization applicable to an OFDM system, and more particularly, to an apparatus and method for OFDM channel equalization, which can prevent divergence, increase a convergence rate, and enhance mean square error (MSE) performance in an equalization scheme using Jacobi iteration applied to an OFDM system in fast time-varying fading channel environments.
2. Description of the Related Art
In fast time-varying fading channel environments, the impulse response of a channel may vary even within one symbol of an orthogonal frequency division multiplexing (OFDM) signal. Thus, interference between sub-carriers in a frequency domain after a fast Fourier transform (FFT), may destroy the orthogonality between the sub-carriers. This phenomenon is called inter carrier interference (ICI).
ICI is caused by adjacent sub-carriers in most cases, and results in serious errors in the processes of channel estimation and compensation. For this reason, a simple single-tap equalization scheme has limitations in enhancing bit error rate (BER) performance.
Thus, considering the rate of channel variations over time in high-mobility environments, equalizers need to be designed with regard to the influences on channels of adjacent sub-carrier symbols, as well as sub-carrier symbols of a corresponding frequency domain.
If the FFT size of an OFDM system is designated as N, a frequency-domain equalization scheme for high-mobility environments is configured so as to obtain the solution of a system of equations with respect to an N×N channel matrix.
However, this scheme of obtaining the solution of a system of equations of the full tap results in excessive complexity. Therefore, the solution of the system of equations may be obtained after a band matrix having a size smaller than N×N is formed.
However, the equalization scheme using the band matrix has a limited performance since the impulse response of a channel exceeds the size of the band matrix in high mobility environments.
To overcome the aforementioned limitations, an equalization scheme using Jacobi iteration (hereinafter, also referred to as Jacobi iteration equalization scheme), one type of iteration scheme using a full matrix, is adopted for the equalization of OFDM systems in fast time-varying fading channel environments. Here, the Jacobi iteration equalization scheme may achieve a significant reduction in the complexity of an inverse matrix.
The Jacobi iteration equalization scheme used for the channel equalization of OFDM systems will now be briefly described.
A transmission signal (x[n]) of an OFDM system may be represented by Equation 1 below:
                              x          ⁡                      [            n            ]                          =                              1            N                    ⁢                                    ∑                              k                =                0                                            N                -                1                                      ⁢                                          X                ⁡                                  [                  k                  ]                                            ⁢                              ⅇ                                  j                  ⁢                                                                          ⁢                  2                  ⁢                                                                          ⁢                  π                  ⁢                                                                          ⁢                                      kn                    /                    N                                                                                                          (        1        )            
where N denotes the size of an FFT, x[n] denotes a transmission signal, X[k] denotes a tap location in a plurality of FFT-processed transmission signals, and ej2πkn/N denotes the phase component of the transmission signal.
A time-domain reception signal, excluding noise, in multipath environments may be represented by Equation 2 below, a signal, after the FFT processing of an OFDM receiver, may be represented by Equation 3 below, and this signal may also be represented in the form of a matrix as shown in Equation 4:
                                              ⁢                              y            ⁡                          [              n              ]                                =                                    ∑                              l                =                0                                            L                -                1                                      ⁢                                                            h                                      l                    ,                    m                                                  ⁡                                  [                  n                  ]                                            ⁢                                                x                  m                                ⁡                                  [                                      n                    -                    1                                    ]                                                                                        (        2        )                                                      y            m                    ⁡                      [            k            ]                          =                              F            ⁢                          {                                                y                  m                                ⁡                                  [                  n                  ]                                            }                                =                                    1              N                        ⁢                                          ∑                                  l                  =                  0                                                  L                  -                  1                                            ⁢                              (                                                      ∑                                          p                      =                      0                                                              L                      -                      1                                                        ⁢                                                                                    X                        m                                            ⁡                                              [                        p                        ]                                                              ⁢                                                                  H                                                  l                          ,                          m                                                                    ⁡                                              [                                                                              (                                                          k                              -                              p                                                        )                                                    N                                                ]                                                              ⁢                                          ⅇ                                                                        -                          j                                                ⁢                                                                                                  ⁢                        2                        ⁢                                                                                                  ⁢                        π                        ⁢                                                                                                  ⁢                                                                              pT                            1                                                    /                          N                                                                                                                                                                            (        3        )                                          [                                                                      Y                  ⁡                                      [                    0                    ]                                                                                                                        Y                  ⁡                                      [                    1                    ]                                                                                                                        Y                  ⁡                                      [                    2                    ]                                                                                                      ⋮                                                                                      Y                  ⁡                                      [                                          N                      -                      3                                        ]                                                                                                                        Y                  ⁡                                      [                                          N                      -                      2                                        ]                                                                                                                        Y                  ⁡                                      [                                          N                      -                      1                                        ]                                                                                ]                =                                                       [                                                          ⁢                                                                                          a                                              0                        ,                        0                                                                                                                        a                                              0                        ,                        1                                                                                                  0                                                        0                                                        …                                                        0                                                                              a                                              0                        ,                                                  N                          -                          1                                                                                                                                                                                a                                              1                        ,                        0                                                                                                                        a                                              1                        ,                        1                                                                                                                        a                                              0                        ,                        1                                                                                                  0                                                        0                                                                                                                                                          0                                                                                        0                                                                              a                                              2                        ,                        1                                                                                                                        a                                              2                        ,                        2                                                                                                                        a                                              2                        ,                        3                                                                                                  0                                                        ⋱                                                        0                                                                                        ⋮                                                        ⋮                                                        ⋮                                                        ⋮                                                                                                                                                          ⋱                                                        ⋮                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                ⋱                                                                                                                                                                                                                                                                                            0                                                                                                                                                          0                                                        0                                                                              a                                                                        N                          -                          2                                                ,                                                  N                          -                          3                                                                                                                                                a                                                                        N                          -                          2                                                ,                                                  N                          -                          3                                                                                                                                                a                                                                        N                          =                          2                                                ,                                                  N                          -                          1                                                                                                                                                                                a                                                                        N                          -                          1                                                ,                        0                                                                                                  0                                                        …                                                        0                                                        0                                                        0                                                                              a                                                                        N                          -                          1                                                ,                                                  N                          -                          1                                                                                                                                ⁢                                                          ]                        =                                                         [                                                                                                    X                        ⁡                                                  [                          0                          ]                                                                                                                                                                        X                        ⁡                                                  [                          1                          ]                                                                                                                                                                        X                        ⁡                                                  [                          2                          ]                                                                                                                                                ⋮                                                                                                                          X                        ⁡                                                  [                                                      N                            -                            3                                                    ]                                                                                                                                                                        X                        ⁡                                                  [                                                      N                            -                            2                                                    ]                                                                                                                                                                        X                        ⁡                                                  [                                                      N                            -                            1                                                    ]                                                                                                                    ]                                                                        (        4        )            
In Equation 4 above, ak,p may be represented by Equation 5 below:
                              a                      k            ,            p                          =                              1            N                    ⁢                                    ∑                              l                =                0                                            L                -                1                                      ⁢                                                            H                                      l                    ,                    m                                                  ⁡                                  [                                                            (                                              k                        -                        p                                            )                                        N                                    ]                                            ⁢                              ⅇ                                                      -                    j                                    ⁢                                                                          ⁢                  2                  ⁢                                                                          ⁢                  π                  ⁢                                                                          ⁢                                                            pT                      1                                        /                    N                                                                                                          (        5        )            
In Equation 5, a channel matrix Hl,m[(k−p)N] is equal to F{hl,m[n]}, and namely, Hl,m[(k−p)N]=F{hl,m[n]}. K denotes the row of a channel matrix, and p denotes the column of the channel matrix.
FIG. 1 is a graph depicting the power of a channel matrix in OFDM channel equalization.
Referring to FIG. 1 and Equation 4, it can be seen that the power of a channel matrix ‘H’ is in the form in which relatively greater values are distributed at a diagonal portion. The Jacobi iteration scheme may be used in the equalization of an OFDM system, which uses this channel matrix configuration.
The channel matrix ‘H’ may include a diagonal matrix ‘D’ and an off-diagonal matrix ‘H−D’ as represented by Equation 6 below:
                              D          =                      [                                                                                a                                          0                      ,                      0                                                                                        0                                                  …                                                  0                                                                              0                                                                      a                                          1                      ,                      1                                                                                        …                                                  0                                                                              ⋮                                                  ⋮                                                  ⋱                                                  ⋮                                                                              0                                                  0                                                  …                                                                      a                                                                  N                        -                        1                                            ,                                              N                        -                        1                                                                                                                  ]                          ,                                  ⁢                              H            -            D                    =                      [                                                            0                                                                      u                                          0                      ,                      1                                                                                        …                                                                      u                                          0                      ,                                              N                        -                        1                                                                                                                                                              ℓ                                          1                      ,                      0                                                                                        0                                                  …                                                                      u                                          1                      ,                                              N                        -                        1                                                                                                                                          ⋮                                                  ⋮                                                  ⋱                                                  ⋮                                                                                                  ℓ                                                                  N                        -                        1                                            ,                      0                                                                                                            ℓ                                                                  N                        -                        1                                            ,                      1                                                                                        …                                                  0                                                      ]                                              (        6        )            
The Jacobi iteration equalization scheme using Equation 4 and Equation 6 may be represented by Equation 7 below:X(r)=X(o)−M·X(r-1), M=D−1(H−D)  (7)where r=1, 2, 3, . . . , which denotes the number of iterations, X(0) denotes an initial estimate value, which is represented by Equation 8 below, M denotes an interference cancellation matrix, D denotes a diagonal matrix, and H denotes a channel matrix.
                              X                      (            0            )                          =                              (                                                            Y                  ⁡                                      [                    0                    ]                                                                    a                                      0                    ,                    0                                                              ,                                                Y                  ⁡                                      [                    1                    ]                                                                    a                                      1                    ,                    1                                                              ,                                                                    Y                    ⁡                                          [                      2                      ]                                                                            a                                          2                      ,                      2                                                                      ⁢                                                                  ⁢                …                            ⁢                                                          ,                                                Y                  ⁡                                      [                                          N                      -                      1                                        ]                                                                    a                                                            N                      -                      1                                        ,                                          N                      -                      1                                                                                            )                    T                                    (        8        )            
However, the related art OFDM channel equalization has the following limitations, and these will now be described with reference to FIG. 2.
FIG. 2 is a graph depicting mean square error (MSE) performance by the related art method for OFDM channel equalization. FIG. 2 shows the result of a simulation on the related art Jacobi iteration equalization scheme in the environment where a normalized Doppler frequency (fd) is ‘0.1’. It can be seen from FIG. 2 that if a0,0, a1,1, . . . , aN-1 and aN-1,N-1, which are the denominators of Equation 8, approach zero, the initial estimate value (X(0)) diverges. In this case, channel equalization cannot be performed normally.